nLab
tangent Lie algebra

Given an algebraic groupGG in characteristic zero, or a finite dimensional Lie group, one associates to it a Lie algebra, its tangent Lie algebra which is the Lie subalgebra 𝒳linv(G)\mathcal{X}^{linv}(G) of the Lie algebra 𝒳(G)\mathcal{X}^(G) left invariant vector fields on GG with respect to the usual Lie bracket of vector fields.

The value of a left-invariant vector field XX at the unit element ee is a tangent vector XeX_e at ee. It appears that the specialization/evaluation at the unit element map 𝒳linv(G)→TeG\mathcal{X}^{linv}(G)\to T_e G is an isomorphism of vector spaces, which is often considered as an identification. However, one needs to look into vector fields in order to find the bracket, hence defining the tangent Lie algebra as the tangent vector space at ee misses the bracket (which come from consideration of infintesimals of second order). One can instead work with right invariant vector fields 𝒳rinv(G)\mathcal{X}^{rinv}(G) and obtain an isomorphic Lie algebra; the isomorphism is of course, by comparing the specialization at ee.

Within 𝒳(G)\mathcal{X}^(G), all right invariant vector fields commute with all left invariant vector fields.

The correspondence G↦𝒳linv(G)G\mapsto\mathcal{X}^{linv}(G) is functorial.